The Gauss Map and the Third Laplace-Beltrami Operator of the Rotational Hypersurface in 4-Space

Abstract

We study and examine the rotational hypersurface and its Gauss map in Euclidean four-space E 4 . We calculate the Gauss map, the mean curvature and the Gaussian curvature of the rotational hypersurface and obtain some results. Then, we introduce the third Laplace–Beltrami operator. Moreover, we calculate the third Laplace–Beltrami operator of the rotational hypersurface in E 4 . We also draw some figures of the rotational hypersurface.

Keywords

• four-space
• rotational hypersurface
• Gauss map
• Gaussian curvature
• mean curvature
• the third Laplace–Beltrami operator